reserve X for ComplexUnitarySpace;
reserve x, y, w, g, g1, g2 for Point of X;
reserve z for Complex;
reserve p, q, r, M, M1, M2 for Real;
reserve seq, seq1, seq2, seq3 for sequence of X;
reserve k,n,m for Nat;
reserve Nseq for increasing sequence of NAT;

theorem
  (z * seq) ^\k = z * (seq ^\k)
proof
  now
    let n be Element of NAT;
    thus ((z * seq) ^\k).n = (z * seq).(n + k) by NAT_1:def 3
      .= z * (seq.(n + k)) by CLVECT_1:def 14
      .= z * ((seq ^\k).n) by NAT_1:def 3
      .= (z * (seq ^\k)).n by CLVECT_1:def 14;
  end;
  hence thesis by FUNCT_2:63;
end;
